What's the difference between the GeometryService areasAndLengths method and the Measurement widget to make a caculation in the ArcGIS JS API? Accuracy?
I don't think these are comparable.
The Areas and Lengths Operation on a Geometry Service, is a web service that can be used to get the Areas and Lengths of an Arbitrary set of polygons in any arbitrary spatial references.
The Measurement widget is a dojo widget that is used in the ESRI JSAPI to draw shapes on the map, and get the area/length etc. If the Map is not in Web Mercator or geographic projection, it will need a geometry service to project or simplify geometries.